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Source file src/math/atan.go

  // Copyright 2009 The Go Authors. All rights reserved.
  // Use of this source code is governed by a BSD-style
  // license that can be found in the LICENSE file.
  
  package math
  
  /*
  	Floating-point arctangent.
  */
  
  // The original C code, the long comment, and the constants below were
  // from http://netlib.sandia.gov/cephes/cmath/atan.c, available from
  // http://www.netlib.org/cephes/cmath.tgz.
  // The go code is a version of the original C.
  //
  // atan.c
  // Inverse circular tangent (arctangent)
  //
  // SYNOPSIS:
  // double x, y, atan();
  // y = atan( x );
  //
  // DESCRIPTION:
  // Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
  //
  // Range reduction is from three intervals into the interval from zero to 0.66.
  // The approximant uses a rational function of degree 4/5 of the form
  // x + x**3 P(x)/Q(x).
  //
  // ACCURACY:
  //                      Relative error:
  // arithmetic   domain    # trials  peak     rms
  //    DEC       -10, 10   50000     2.4e-17  8.3e-18
  //    IEEE      -10, 10   10^6      1.8e-16  5.0e-17
  //
  // Cephes Math Library Release 2.8:  June, 2000
  // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
  //
  // The readme file at http://netlib.sandia.gov/cephes/ says:
  //    Some software in this archive may be from the book _Methods and
  // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
  // International, 1989) or from the Cephes Mathematical Library, a
  // commercial product. In either event, it is copyrighted by the author.
  // What you see here may be used freely but it comes with no support or
  // guarantee.
  //
  //   The two known misprints in the book are repaired here in the
  // source listings for the gamma function and the incomplete beta
  // integral.
  //
  //   Stephen L. Moshier
  //   moshier@na-net.ornl.gov
  
  // xatan evaluates a series valid in the range [0, 0.66].
  func xatan(x float64) float64 {
  	const (
  		P0 = -8.750608600031904122785e-01
  		P1 = -1.615753718733365076637e+01
  		P2 = -7.500855792314704667340e+01
  		P3 = -1.228866684490136173410e+02
  		P4 = -6.485021904942025371773e+01
  		Q0 = +2.485846490142306297962e+01
  		Q1 = +1.650270098316988542046e+02
  		Q2 = +4.328810604912902668951e+02
  		Q3 = +4.853903996359136964868e+02
  		Q4 = +1.945506571482613964425e+02
  	)
  	z := x * x
  	z = z * ((((P0*z+P1)*z+P2)*z+P3)*z + P4) / (((((z+Q0)*z+Q1)*z+Q2)*z+Q3)*z + Q4)
  	z = x*z + x
  	return z
  }
  
  // satan reduces its argument (known to be positive)
  // to the range [0, 0.66] and calls xatan.
  func satan(x float64) float64 {
  	const (
  		Morebits = 6.123233995736765886130e-17 // pi/2 = PIO2 + Morebits
  		Tan3pio8 = 2.41421356237309504880      // tan(3*pi/8)
  	)
  	if x <= 0.66 {
  		return xatan(x)
  	}
  	if x > Tan3pio8 {
  		return Pi/2 - xatan(1/x) + Morebits
  	}
  	return Pi/4 + xatan((x-1)/(x+1)) + 0.5*Morebits
  }
  
  // Atan returns the arctangent, in radians, of x.
  //
  // Special cases are:
  //      Atan(±0) = ±0
  //      Atan(±Inf) = ±Pi/2
  func Atan(x float64) float64
  
  func atan(x float64) float64 {
  	if x == 0 {
  		return x
  	}
  	if x > 0 {
  		return satan(x)
  	}
  	return -satan(-x)
  }
  

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